An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 41-43.

Examples.

71

whose higher powers, when n increases, have zero as limit. In
the present case therefore the first and second hypotheses are comprehended in one: A function whose n derived functions up to n = o
remain finite within an interval, call be calculated in this interval
by a series of powers. But this proposition cannot be converted, because Lim B can vanish, without the n derived functions up to n = oo
also being finite, as some of the following examples show.
42. Exponential series: y == f x) = ex.
For x = 0, f (x) and all derived functions are known; in fact fn (x) ex
therefore fn (0) = 1. These are continuous functions for all finite
values of x, and even for n = oo always remain finite. Accordingly
Taylor's series converges and
X x2 33 xk
ex - L -. + -. in infin., - <oo < x < +- o.*)
If more generally y =- a (a > 0), let us put y - = exl- and we have
xla (xIla) ( cxla) (xla )
I, 1 12 + 12 + - +I * * - I +.. infin.
- 00 < X < + 00.
43. Trigonometric series: y=- fx) == sin x.
dY = sin (x + - in z) is finite for every finite argument. Accordingly
d x'1
Taylor's series converges, and as
f(0) = sin (0) - 0, f~I (0) = sin (- n ) - 0,
f'(O)- sin () 1) =, (0) =sin (3 t) =, fv (0) - sin () = 1,
we have:
- yi S, 3 iy.5-,x7 S S 2A+
siln x - -.. 0 <X <+ o.,
Again: y = f(x) = cos x. CdY- =os (x + Init).
f(0) c () /S ( 0) = s ( ) - 1, I (0) - ) 1 V = cos (A ) ) 1,
/"(0)= cos (. it)=0, f"'(O) = cos (i- ) 0=, / (0) = cos (- i7) — 0,
cos =- 1 - +E —'" (- )-*'o - oo < x < + o0.
The present series render it possible to calculate trigonometric tables
for the sine and cosine of any number. When we wish to abstract
quite from the geometric definition of sine and cosine, these series
*) The series itself was first established by Newton, as well as the series
for sine and cosine; the number e, as already mentioned, was introduced by
Euler as basis of exponential functions.